arXiv subjects

Ai Huang

Publications and source records attributed to Ai Huang.

3 recordsLinked to original sources

arXiv2026

Spatial inhomogeneity for a three-dimensional doubly degenerate nutrient system with indirect consumption

This paper investigates the global dynamics of a doubly degenerate nutrient-taxis system with indirect consumption: \begin{equation*} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\nabla \cdot (u^{2}v\nabla v)+\ell vw,&x\in \Omega,\, t>0,\\ & v_{t}=\Delta v-vw,&x\in \Omega,\, t>0,\\ &w_t=\Delta w-w+u,&x\in\Omega,t>0 \end{aligned} \right. \end{equation*} posed on a smooth bounded domain $\Omega\subset\mathbb{R}^{3}$ with no-flux boundary conditions. It is shown that for suitably regular initial data $(u_0,v_0,w_0)$, the associated initial-boundary value problem admits a global weak solution. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0,w_\infty)$ as $t\rightarrow \infty$. Notably, when $u_0$ is nonconstant and the mass of $v_0$ is sufficiently small, the limiting profiles $u_{\infty}$ and $w_{\infty}$ are are spatially nonhomogeneous, capturing emergent patterning in nutrient-depleted environments. A cornerstone of our analysis is the introduction of novel functional inequalities, which provide estimates from below for the integral $\int_{\Omega}u^{k}v|\nabla u|^2$ with some $k>-1$.

math.AP↗

arXiv2026

Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system

The doubly degenerate nutrient taxis system \begin{equation}\label {0.1} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha}v\nabla v)+\ell uv,&x\in \Omega,\, t>0,\\ & v_{t}=\Delta v-uv,&x\in \Omega,\, t>0,\\ \end{aligned} \right. \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^3$ where $\alpha>0,\chi>0$ and $\ell> 0$. By developing a novel class of functional inequalities to address the challenges posed by the doubly degenerate diffusion mechanism in \eqref{0.1}, it is shown that for $\alpha\in(\frac{3}{2},\frac{19}{12})$, the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0)$ as $t\rightarrow \infty$. Notably, the limiting profile $u_{\infty}$ is non-homogeneous when the initial signal concentration $v_0$ is sufficiently small, provided the initial data $u_0$ is not identically constant.

math.AP↗

arXiv2025

Small-mass solutions in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with indirect nutrient consumption

This paper is concerned with the singular chemotaxis-fluid system with indirect nutrient consumption: $ n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(n S(x,n,v)\cdot \nabla v);\ v_{t}+u\cdot\nabla v=\Delta v-vw;\ w_{t}+u\cdot\nabla w=\Delta w-w+n;\ u_t+(u\cdot\nabla) u=\Delta u-\nabla P+n\nabla\Phi;\ \nabla\cdot u=0\ $ in a smooth bounded domain $\Omega\subset\mathbb{R}^2$ under no-flux/Neumann/Neumann/Dirichlet boundary conditions, where $\Phi\in W^{2,\infty}(\Omega)$, and $S: \overline{\Omega}\times [0,\infty) \times (0,\infty)\rightarrow\mathbb{R}^{2\times 2}$ is a suitably smooth function that satisfies $|S(x,n,v)|\leq S_0(v) /v $ for all $(x,n,v) \in \Omega\times (0,\infty)^2$ with some nondecreasing $S_0: (0,\infty)\rightarrow(0,\infty)$. For all reasonably regular initial data with a smallness assumption merely involving the quantity $\int_\Omega n_0$, it is shown that the problem possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state $( \frac{1}{|\Omega|}\int_{\Omega}n_0,0,\frac{1}{|\Omega|}\int_{\Omega}n_0,0)$ with respect to the norm in $L^\infty(\Omega)$. This rigorously confirms that, at least in the two-dimensional setting, in comparison to the direct mechanism of nutrient consumption, an indirect mechanism can induce much more regularity of solutions to the chemotaxis--fluid system even with a singular tensor-valued sensitivity.

math.AP↗